Selecta Mathematica

, Volume 23, Issue 1, pp 389–423 | Cite as

Derived categories of cyclic covers and their branch divisors

Article

Abstract

Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover \(X \rightarrow Y\) ramified over a divisor \(Z \subset Y\). We construct semiorthogonal decompositions of \(\mathrm {D^b}(X)\) and \(\mathrm {D^b}(Z)\) with distinguished components \({\mathcal {A}}_X\) and \({\mathcal {A}}_Z\) and prove the equivariant category of \({\mathcal {A}}_X\) (with respect to an action of the nth roots of unity) admits a semiorthogonal decomposition into \(n-1\) copies of \({\mathcal {A}}_Z\). As examples, we consider quartic double solids, Gushel–Mukai varieties, and cyclic cubic hypersurfaces.

Mathematics Subject Classification

14F05 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Algebraic GeometrySteklov Mathematical InstituteMoscowRussian Federation
  2. 2.The Poncelet LaboratoryIndependent University of MoscowMoscowRussian Federation
  3. 3.Laboratory of Algebraic Geometry and its ApplicationsNational Research University Higher School of EconomicsMoscowRussian Federation
  4. 4.Department of MathematicsHarvard UniversityCambridgeUSA

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